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Full text of "Treatise On Analysis Vol-Ii"

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16 BAiRE'S THEOREM AND ITS CONSEQUENCES 91 Let x0 e E. We have to show that, for each neighborhood V of e in G, the set V • *0 is a neighborhood of *0 in E (12.11.5). Let W be a compact symmetric neighborhood of e in G such that W2 c: V, and let (sn) be a sequence which is dense in G, such that (jflW) is a covering of G. Each of the sets snVi • x0 is closed in E, because s^s- x0 is continuous (12.3.6), and E is the union of the denumerable sequence of closed sets sn W • x0. By Baire's theorem (12.16.1), there exists an index n such that 5-nW • x0 has an interior point sns- x0, with s e W. It follows (12.10.3) that x0 is an interior point of the set in other words, V • x0 is a neighborhood of x0. Q.E.D. (12.16.13) Let Gbea separable, metrizable, locally compact group, let G' be a metrizable group, and letf: G -* G' be a continuous surjective homomorphism. Then f is a strict morphism (12.12.7) (in other words, if H is the kernel of/ then the canonical bijection g: G/H -> G' is an isomorphism of topological groups). For we may consider G' as a space on which G acts continuously and transitively by the rule (s9 t')i-*f(s)t', and the stabilizer of the neutral element e1 of G' is H. The result therefore follows from (12.16.12). PROBLEMS 1. Let E, F be two metric spaces, A a dense subspace of E, and/a continuous mapping of A into F. If F is complete, show that the set of points in E at which/has no limit relative to A (3.13) is meager in E. (For each n, consider the set of points xe E at which the oscillation of/with respect to A (3.14) is >!//?.) 2. Let E, F be two complete metric spaces, and/a homeomorphism of a dense subspace A of E onto a dense subspace B of F. Show that there exists a subspace C ^> A in E (resp. D ^ B in F) which is a denumerable intersection of open sets, and an extension of/to a homeomorphism g of C onto D. (Apply Problem 1 to/and its inverse.) 3. Let E be a complete metric space, F a metric space, and (/„) a sequence of continuous mappings of E into F which converges simply (pointwise) in E to a mapping/. Show that the set of points x e E at which /is not continuous is meager in E. (Let Gp,n be the set of x e E for which the distance between fp(x) and f^(x) is <l/2n for all q >p. Show that the union of the interiors of the sets Gp, „ for p 2> 1 is a dense open set, by using Baire's theorem. Deduce that the set of points at which the oscillation of /is ^l/n contains a dense open set.) Generalize to the case where it is assumed only that, for each /?, the restriction of fn to the complement of a meager set M« (depending on n) is continuous,t, for each contraction U e tf and each vector x e E, the